Mesh with shape key animation - ok in Blender, wrong in Unity
To fix the scaling issue, be sure you select your mesh in Object mode and check the scale transform. If it's not set to 1.0, apply your scale by pressing CtrlA and selecting Scale from the drop down menu that appears. If you're using an armature, make sure your mesh and armature have their scales applied before binding them together. 

To fix the rotation issue, check that any rotation changes have also been applied, similarly to how you apply scaling changes. If you're exporting with FBX, change your Up and Forward directions in the FBX export screen to the correct directions for your mesh.